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What is the right way to think of omitted variable bias in a regression that only has dummy variables?

Let's say I have the following equation:

(1) y=β0+β1x1+β2x2+β3x3+ϵ,

where y is the price of shoes; and x1, x2 and x3 are dummy variables for three different regions (x4, the reference region was omitted).

And I suspect that I am missing a variable (x4) to adjust for 'level of urbanization of each region' -- each region contains an uneven number of areas with different levels of urbanization. Thus, the true model is, or so I suspect:

(2) y=β0+β1x1+β2x2+β3x3+δx4+ϵ

Now I know I can sign the bias of any one of the coefficients in equation (1) if two conditions are met: a) x4 is correlated with either x1, x2 or x3; and b) x4 has an impact on y (i.e. δ>0).

However, I am not sure if I can explore condition a) above since the correlation of x4, in this case, would be with a categorical variable, not a continuous one.

How can I go about this?

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  • $\begingroup$ How do you measure "level of urbanization in each city" (i.e. in three cities) with one single variable? If this is the level of urbanization in the $i$th city (with $y_i$ being the response variable) then this would be perfectly collinear with x1-x3, i.e., the dummy variables for the city. $\endgroup$ – Tamas Ferenci Jun 12 at 17:48
  • $\begingroup$ Sorry. I meant to say 'level of urbanization within areas of each region'. Each region has an uneven number of areas and the urbanization variable varies from areas to area within regions. The question has been edited. $\endgroup$ – StatsScared Jun 12 at 18:23
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You're right that the requirement is $\mathrm{cov}\left(x_4,x_1\right)\neq0$. The important part is that $\mathrm{cov}$ doesn't care if any of the variables (or both) is continuous or categorical. You can calculate it, irrespectively of what is their nature!

Concretely in your case: if $x_1$ is binary, and $x_4$ is continuous then

$\mathrm{cov}\left(x_4,x_1\right)=\frac{1}{n}\sum_{i=1}^n \left(x_{1i}-\overline{x_1}\right)\left(x_{4i}-\overline{x_4}\right),$

i.e. the definition is totally the same. What you have to note in this formula, is that you can calculate everything here, it doesn't matter that $x_1$ is binary!

Yes, the formula will become

$\mathrm{cov}\left(x_4,x_1\right)=\frac{1}{n}\sum_{i=1}^n \left(x_{1i}-\overline{x_1}\right)\left(x_{4i}-\overline{x_4}\right)=\frac{-\overline{x_1}\sum_{x_{1i}=0}\left(x_{4i}-\overline{x_4}\right) + \left(1-\overline{x_1}\right)\sum_{x_{1i}=1}\left(x_{4i}-\overline{x_4}\right)}{n},$

but that is just a technical question (simplification).

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  • $\begingroup$ How exactly do I interpret cov(x4,x1) though? Covariance between levels of urbanization with ... ? $\endgroup$ – StatsScared Jun 12 at 19:00
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    $\begingroup$ ...with being in the first region. I.e., how strongly being in the first region determines the level of urbanization. For instance, if the distribution of $x_4$ is the same in region 1 and outside region 1, then both sums will be zero (as $\overline{x_4}$ will also be the group mean in the $x_1=0$ and $x_1=1$ groups), so the covariance is zero. $\endgroup$ – Tamas Ferenci Jun 12 at 19:37
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    $\begingroup$ At the other extreme, if every urbanization is the same ($c_1$) in region 1 without variation, and same ($c_0$) outside without variation (but these two values are not equal, of course, $c_1\neq c_0$), then the covariance will be the product of the two standard deviation, so the correlation will be 1. $\endgroup$ – Tamas Ferenci Jun 12 at 19:44
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Whether a variable is binary or continuous has no bearing on its potential for omitted variable bias (unless it's a poorly measured version of a truly continuous variable). If the "true" model uses the binary variables as they are, then the same rules that apply for continuous predictors and continuous omitted variables apply here. Regression models don't know whether predictors are continuous or binary; they treats them all as continuous. It is only our interpretation of them as binary that makes them any different from continuous variables.

Again, things get hairy if the binary variable is a categorized version of a continuous variable (or has measurement error in it some other way).

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  • $\begingroup$ Thanks. How should I think about the relationship between a dummy regional value and, for example, a) a continuous variable measuring the level of urbanization within areas of each region, and b) a case like the one you say is difficult, if my urbanization variable is split into two binary variables (for example, if urbanization rate is >40% in one case, and equal or above that threshold in another) $\endgroup$ – StatsScared Jun 12 at 18:33
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    $\begingroup$ For the first case, the omitted variable bias formulas will be the same. If you have perfectly measured variables, it doesn't matter what the variable types are. The omitted variable bias formulas are agnostic to the variable types. I don't know enough about omitted variable bias in the context of measurement error to respond adequately to the second case. $\endgroup$ – Noah Jun 12 at 18:37
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    $\begingroup$ @StatsScared using dummies to adjust for region is actually more powerful than including a single continuous variable at the region level. This is because a) urbanization is but one way that a region contributes to the likelihood of response and b) the relationship between urbanization and response could be non-linear. $\endgroup$ – AdamO Jun 12 at 18:49
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    $\begingroup$ @StatsScared no. I suggested a dummy for each region. A dummy is a way of representing a categorical variable. If you threshold a continuous variable, e.g. based on a 40% cut as you wrote above, you will certainly lose a lot of power. That's called dichotomania. $\endgroup$ – AdamO Jun 12 at 19:35
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    $\begingroup$ @StatsScared originally you spoke of cities and of regions separately, so the content of this question is a moving target. $\endgroup$ – AdamO Jun 13 at 21:22
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If we focus on the primary purpose of a regression model as summarizing mean differences, it's easy to see there's no such thing as an omitted variable bias. In fact, some causal schools of thought (such as the one to which I belong) think that so long as there is any residual standard error, then variables have been omitted (no such thing as inherent randomness). All "omitted variable bias" amounts to is an imprecise catch-all which lumps together confounding and subpar predictions.

If your conditional mean response is summarized by 3 fixed effects for the four-level region variable, what you have is a simple kind of ANOVA. If you add the between-region adjustment variable of urbanization, you have a special type of case-mix-adjusted model. Some theory shows that the added adjustment changes the interpretation of the fixed effects for region by standardizing for urbanization. What you get is a summary of mean differences between region that is independent of urbanization. How you interpret that depends on how urbanization is calculated. This enables thought-experiment comparisons such as what would be the expected response in region A if it had the level of urbanization of region B? (be careful because a lot of thought experiments are useless if they are not part of a prespecified analysis plan).

So no you haven't omitted any variable, and your result is not biased, you just have a model that summarizes all available information. I would reserve a discussion about "urbanization" to contextualize the different ways it could be measured, what is known about the regions, and implications for further research.

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